The Faber polynomials for \(m\)-fold symmetric domains
Let \(E\) be a compact continuum in \(\mathbb{C}\), and let \(z= \psi(w)= w+ b_ 0+ {b_ 1\over w}+ {b_ 2\over w^ 2}+\cdots\) be the normalized conformal map from \(\{w: | w|> \rho\}\) onto the complement of \(E\). The Faber polynomials \(F_ n\) associated with \(E\) can be computed recursively if the \(b_ j\) are known. In the case that \(E\) is bounded by \(\{z\in \mathbb{C}: | z^ m- 1|= 1\}\) \((m= 2,3,\dots)\) or bounded by \(\{z\in \mathbb{C}: z= (w^ 2+ w+ 1)^{1/2}, | w|= 1\}\), the author gives explicit expressions for the \(F_ n\). Noteworthy is also the representation \(F_ n(z)= \text{det}(zI_ n- H_ n)\), where \(I_ n\) is the \(n\times n\) unit matrix, and where \(H_ n\) is an \(n\times n\) matrix in which the \(b_ j\) enter. The zeros of \(F_ n\) are therefore the eigenvalues of \(H_ n\). In the examples studied, these zeros are dense on the lemniscates.
scientific article; zbMATH DE number 739562
scientific article